The Spherical Harmonics Expansion (SHE) assumes a momentum distribution function only depending on the
microscopic kinetic energy. The SHE-Poisson system
describes carrier transport in semiconductors with self-induced electrostatic potential.
Existence of weak solutions to the SHE-Poisson system subject to periodic boundary conditions
is established, based on appropriate a priori estimates and a Schauder fixed point procedure.
The long time behavior of the one-dimensional Dirichlet problem
with well prepared boundary data is studied by an entropy-entropy dissipation method. Strong convergence
to equilibrium is proven. In contrast to earlier work, the analysis is carried out without the use of the
derivation from a kinetic problem.